_{ Find the value of c for which $\displaystyle \int^1_0 x^3sinh(cx) dx = 1 $ I have no idead on how to work this out }
You have a product of two functions so the first thing I would think of is "itegration by parts" which is the reverse of the "product rule" for derivatives.
Let $\displaystyle u= x^3$ and $\displaystyle dv= sinh(cx)dx$ so that $\displaystyle du= 3x^2dx$ and $\displaystyle v= \frac{1}{c}cosh(cx)$.
Do you see how the degree of x has been reduced? Now do that two more times.